1. Let p, q, r be the propositions,

Prove the following propositional statements using propositional rules. (photo attached)

a. show that p v q v ~p is a tautology (p or q or not p)

b. show that (p ^ ~q) ^ (~p ^ q) is a contadiction

c. show that p q and ~q --> ~p are logically equivalent

2. Let p = employee is a manager

q = employee can charge overtime

r = employee can vote

Express using a combination of p, q, r the following prepositional statements

a. managers can charge overtime

b. only managers can vote

c. voting employees can charge overtime

d. Determine if the system is consistent or inconsistent using a truth table

p | q | r | value

TABLE 6 Logical Equivalences. TABLE 2 De Morgan's Laws. Equivalence Name ldentity laws PAF-F Idempotent laws Double negation law PA A Associative laws p v (q ^ r) (p v q) (p Vr) Distributive laws De Morgan's laws Absorption laws Negation laws TABLE 7 Logical Equivalences Involving Conditional Statements TABLE8 Logical Equivalences Involving Biconditional Statements